Theorem seqeq1 10907

Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)

Assertion

Ref	Expression
seqeq1	|- ( M = N -> seq M ( .+ , F ) = seq N ( .+ , F ) )

Proof of Theorem seqeq1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . . . . 6 |- ( M = N -> M = N )
2		fveq2 5698	. . . . . 6 |- ( M = N -> ( F ` M ) = ( F ` N ) )
3	1, 2	opeq12d 3912	. . . . 5 |- ( M = N -> <. M , ( F ` M ) >. = <. N , ( F ` N ) >. )
4		freceq2 6669	. . . . 5 |- ( <. M , ( F ` M ) >. = <. N , ( F ` N ) >. -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
5	3, 4	syl 14	. . . 4 |- ( M = N -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
6		fveq2 5698	. . . . . 6 |- ( M = N -> ( ZZ>= ` M ) = ( ZZ>= ` N ) )
7		eqid 2238	. . . . . 6 |- _V = _V
8		mpoeq12 6151	. . . . . 6 |- ( ( ( ZZ>= ` M ) = ( ZZ>= ` N ) /\ _V = _V ) -> ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) )
9	6, 7, 8	sylancl 417	. . . . 5 |- ( M = N -> ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) )
10		freceq1 6668	. . . . 5 |- ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) = ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) = frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
11	9, 10	syl 14	. . . 4 |- ( M = N -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) = frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
12	5, 11	eqtrd 2271	. . 3 |- ( M = N -> frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
13	12	rneqd 5013	. 2 |- ( M = N -> ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. ) = ran frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. ) )
14		df-seqfrec 10905	. 2 |- seq M ( .+ , F ) = ran frec ( ( x e. ( ZZ>= ` M ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. M , ( F ` M ) >. )
15		df-seqfrec 10905	. 2 |- seq N ( .+ , F ) = ran frec ( ( x e. ( ZZ>= ` N ) , y e. _V |-> <. ( x + 1 ) , ( y .+ ( F ` ( x + 1 ) ) ) >. ) , <. N , ( F ` N ) >. )
16	13, 14, 15	3eqtr4g 2296	1 |- ( M = N -> seq M ( .+ , F ) = seq N ( .+ , F ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   _Vcvv 2821   <.cop 3712   ran crn 4777   ` cfv 5380  (class class class)co 6088   e. cmpo 6090  freccfrec 6666   1c1 8186   + caddc 8188   ZZ>=cuz 9936   seqcseq 10904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-cnv 4784  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fv 5388  df-oprab 6092  df-mpo 6093  df-recs 6581  df-frec 6667  df-seqfrec 10905

This theorem is used by:  seqeq1d  10910  seq3f1olemqsum  10970  seqf1oglem2  10977  seq3id  10982  seq3z  10985  iserex  12129  summodclem2  12173  summodc  12174  zsumdc  12175  isumsplit  12282  ntrivcvgap  12339  ntrivcvgap0  12340  prodmodclem2  12368  prodmodc  12369  zproddc  12370  fprodntrivap  12375  ege2le3  12462  gzsumfzval  13769  gzsumval2  13772  logfac  16143